Tracking critical points on evolving curves and surfaces
In recent years it became apparent that geophysical abrasion can be well characterized by the time evolution of the number of static balance points of the abrading particle. Static balance points correspond to the critical points of the particle’s surface represented as a scalar distance function , measured from the center of mass of the particle, so their time evolution can be expressed as . The mathematical model of the particle can be constructed on two scales: on the macro (global) scale the particle may be viewed as a smooth, convex manifold described by the smooth distance function with equilibria, while on the micro (local) scale the particle’s natural model is a finely discretized, convex polyhedral approximation of , with equilibria. There is strong intuitive evidence suggesting that under some particular evolution models (e.g. curvature-driven flows) and primarily evolve in the opposite manner (i.e. if one is increasing then the other is decreasing and vice versa). This observation appear to be a key factor in tracking geophysical abrasion. Here we create the mathematical framework necessary to understand these phenomenon more broadly, regardless of the particular evolution equation. We study micro and macro events in one-parameter families of curves and surfaces, corresponding to bifurcations triggering the jumps in and . Based on this analysis we show that the intuitive picture developed for curvature-driven flows is not only correct, it has universal validity, as long as the evolving surface is smooth. In this case, bifurcations associated with and are coupled to some extent: resonance-like phenomena in can be used to forecast downward jumps in (but not upward jumps). Beyond proving rigorous results for the limit on the nontrivial interplay between singularities in the discrete and continuum approximations we also show that our mathematical model is structurally stable, i.e. it may be verified by computer simulations.
Key words and phrases:equilibrium, convex surface, Poincaré-Hopf formula, polyhedral approximation, curvature-driven evolution.
1991 Mathematics Subject Classification:53A05, 53Z05
- 1 Introduction
- 2 Static theory: equilibria on finely discretized, fixed planar curves
- 3 Dynamic theory: local equilibria on finely discretized, evolving planar curves
- 4 Smooth surfaces and their discretizations
- 5 Discussion and conclusions
- 6 Acknowledgement
Recent work in geomorphology [12, 26, 35] indicates that the shapes of sedimentary particles and the time () evolution of those shapes may be well characterized by the number of mechanical balance points of the abrading particle. Such balance points correspond to the critical points of the scalar distance function , measured from the center of mass . In this paper we develop the mathematical theory for the case when is a smooth, convex planar curve but we will also show numerical results for the non-smooth case and for surfaces.
There are two types of physical models describing the evolution of particles under abrasion. The first kind, which we might call local model, is based on discrete events when in a collision a small part of the abraded particle is broken off. The most natural geometrical setting for local models is a multi-faceted convex polyhedron and collisional events correspond to truncations with planes parallel to, and very close to tangent planes. The second kind of model we might call global and it considers the averaged effect of many such micro collisions. The natural setting for global models is a smooth, convex body evolving under a geometric partial differential equation (PDE). Both model types are physically legitimate: at close inspection, the convex hulls of pebbles can be best approximated by multi-faceted polyhedra, on the other hand, it is equally possible to adopt the global view and approximate pebbles with smooth surfaces.
The distance functions describing these two types of models are, of course, related: in case of a fine polyhedral approximation the two surfaces (smooth global model and polyhedral discrete model) are close to each other in the -norm. We denote the size of the largest polyhedral face by and the two distance functions by and , respectively. Mechanical equilibria correspond to the critical points of and . We will refer to these points as global and local equilibria and denote their numbers by and respectively. Both the smooth function and its polyhedral approximation may carry equilibria of different stability types. In three dimensions we have three generic types: stable, saddle an unstable. For example, vertices of a polyhedron may carry unstable equilibrium points, edges may carry saddle-type equilibrium points and faces may carry stable equilibrium points. The numbers and refer to the number of equilibria belonging to any of the aforementioned stability types. In two dimensions we just have two generic stability types: stable and unstable equilibria follow each other alternating along the smooth curve or its fine polygonal approximation . Figure 1 illustrates these concepts for a planar, elliptical disc and its discretized, polygonal approximation. Although critical points appear to be related to first derivatives (they are defined by vanishing gradient), nevertheless, the -proximity of the two functions does not imply that and are close. As we showed in , in the limit does not, in general, converge to .
The time dependence of the number of global critical points has been broadly investigated in various evolution equations [10, 12, 14, 15, 22, 25]. Our goal here is rather different: instead of studying any particular evolution equation (which we will use only as illustrations) we focus on some universal features relating to . Earlier results appear to suggest intuitively that, as long as is smooth, and tend to evolve in the opposite directions: in  it was shown that it is always possible to increase the number of equilibria via suitable, small truncations, however, the opposite is not true: in general, it is not possible to reduce the number of equilibria by a small local truncation. These results were further advanced in  where the concept of robustness was introduced to measure the stability of with respect to truncations of the solid. By distinguishing between upward and downward robustness (measuring the difficulty to increase or decrease , respectively) it was again found that upward robustness is, in general, much smaller than downward robustness. These results suggest that in the local, polygonal model, which could be realized in a randomized chipping algorithm [20, 24], would tend to increase under subsequent, small random truncations. On the other hand, there are results [22, 12] showing that, at least in curvature-driven, global PDE models which could be regarded as continuum analogies of the aforementioned chipping algorithms, tends to decrease.
In this paper we will show that the indicated opposite trend of and is indeed universal, and it is independent of the particular type of evolution model as long as remains smooth. Our paper will focus on this case, and we will show that resonance-like phenomena in may help to predict downward jumps in . The nontrivial coupling between and may be better understood intuitively via an analogy to a mechanical oscillator. In case of of a damped, driven harmonic oscillator resonance occurs whenever the driving frequency approaches the natural frequency of the oscillator, i.e. an extrinsic quantity approaches and intrinsic one. In our problem we associate two scalars with a point of a smooth curve: the distance between and the center of mass , and the radius of curvature at . In the analogy, is the extrinsic and is the intrinsic quantity. (In three dimensions we have two intrinsic quantities: the two principal radii) The size of the discretization is analogous to damping and is analogous to the amplitude of the oscillation. If then in the limit we can observe as . However, the analogy is incomplete, because in the case of the harmonic oscillator a single amplitude-frequency diagram is sufficient to describe the generic response of the system. In our geometric setting there exist two, distinct generic scenarios which we explain below.
As , the trajectory of the center of mass approaches the evolute or corresponding to and , respectively [17, 31], see also Remarks 5, 6. We will refer to the intersection between the trajectory of and the evolute or as micro and macro events, respectively. It is well-known  that these intersections trigger upward or downward jumps in the integer-valued functions and . In the generic case an event is equivalent to a codimension one, saddle-node bifurcation. Such a bifurcation can occur in two different manners: either creating or annihilating one pair of equilibrium points. As we will show, micro events, in general, do not trigger macro events, however, the opposite is not true: jumps in occur whenever and here we have always . However, the time evolution will depend on thy type of macro-event in an asymmetric manner (cf. Figure 2):
If increases by 2 then we call this a (generic) creation and the corresponding macro event in will be of type .
If decreases by 2 then we call it a (generic) annihilation and the corresponding macro event in will be of type .
The existence of coupled macro-events sheds light on the previous intuition about the opposite co-evolution of and . As we can see in Figure 2(B), if evolves in a monotonic fashion (i.e. it has jumps only in one direction) then, due to the coupling between corresponding macro events, for most of the time the evolution of will be in the opposite direction. This does not imply that will be monotonic in any averaged sense, however, in a generic case, locally it will almost always appear to be monotonic. Figure 2(B) illustrates the qualitative trends for and in the limit. Since the mesh-size is analogous to damping, in computer simulations, for finite values of we expect to see finite versions of -type and -type events as well as small fluctuations of , cf. Figure 4.
Beyond explaining earlier observations, these results can also be of practical use. Both computer simulations of the PDEs describing abrasion processes and related laboratory measurements are inherently discrete, one good example is the study of surfaces of natural pebbles which, while rolling on a horizontal plane, are supported on their convex hull . The latter is well approximated by a many-faceted polyhedron (with faces of maximal size ), on which, by studying its detailed 3D scanned images, one can clearly observe large numbers of adjacent equilibria in strongly localized flocks. The macro events of type and in the evolution of correspond to the explosion of these local flocks into huge critical flocks the size and evolution of which we explore in Subsections 3.1 and 3.2. If we approximate the polyhedron by a sufficiently smooth surface , then, in a generic case, we can see that the flocks of equilibria on appear in the close vicinity of the (isolated) equilibrium points of (cf. Figure 3), however, the latter may not be directly observed on the polyhedral image. In computer simulations the opposite happens: a smooth surface is replaced by its fine discretization and computations are performed on the latter. As we can see, it is often the case that we have means to monitor but we may not be able to directly monitor , although the latter is of prime physical interest . In such cases by using Theorem 3, may be obtained simply via monitoring the - and -type events in and computing as
where and refer respectively to the number of -type and -type events observed in .
These macro events also connect to the aforementioned robustness concepts. The quantity may serve as a measure of downward robustness since whenever , the function is approaching a downward step. Curiously, does not carry advance information on the approaching upward step in . The asymmetry in increasing/decreasing is at the very heart of understanding natural abrasion processes and their mathematical models. One, rather delicate feature of these PDEs is whether they tend to increase, conserve or reduce [22, 12] and the ability to track and measure this phenomenon in experiments and computations is of key importance in the identification and scaling of the proper evolution equations. The dynamic theory for local equilibria, the central topic of this paper, appears to be a necessary step towards this goal. Our paper is structured as follows.
To understand their evolution, the first natural question is to ask for the relationship between and on a fixed surface described by a generic distance function . We addressed this problem (which we may call the static theory of equilibria) in  and we obtained explicit formulae (to be reviewed in Subsection 1.2.2) for the size of the individual flocks (emerging in the limit) surrounding generic critical points of . However, those results do not permit the computation of the global value on the whole surface because in  we did not exclude the existence of equilibria outside flocks. In Subsection 2.1 we complement the static theory by filling this gap; we will prove that local equilibria disconnected from flocks do not exist. In Subsection 2.2 we further strengthen these results by showing the structural stability of our formulae with respect to small random fluctuations in mesh size. The latter result validates computer simulations running with slightly unequal mesh size. The main focus of our current paper is the dynamic theory which we develop in Section 3. We prove our results for planar curves, however, in Section 4 we provide a visually attractive numerical example for the evolution of critical flocks on surfaces.
1.2. Basic concepts
The study of equilibria of rigid bodies was initiated by Archimedes ; his results have been used in naval design even in the 18th century (cf. ). Archimedes’ main concern was the number of the stable balance points of the body.
Mechanical equilibria of convex bodies correspond to the singularities of the gradient vector field characterizing their surface. Modern, global theory of the generic singularities of smooth manifolds appears to start with the papers of Cayley  and Maxwell  who independently introduced topological ideas into this field yielding results on the global number of stationary points. These ideas have been further generalized by Poincaré and Hopf, leading to the Poincaré-Hopf Theorem  on topological invariants. In case of topological spheres in two and three dimensions, the Poincaré-Hopf formula can be written as
where denote the numbers of ‘sinks’ (minima, corresponding to stable equilibria), ‘sources’ (maxima, corresponding to unstable equilibria) and saddles, respectively. This formula can be also regarded as a variant of the well-known Euler’s formula  for convex polyhedra.
Mechanical equilibria of polyhedra have also been investigated in their own right; in particular, the minimal number of equilibria attracted substantial interest. Monostatic polyhedra (i.e. polyhedra with just stable equilibrium point) have been studied in ,,  and .
The total number of equilibria has also been in the focus of research. In planar, homogeneous, convex bodies (rolling along their circumference on a horizontal support), we have . However, convex homogeneous objects with exist in the three-dimensional space (cf. ). Zamfirescu  showed that for typical convex bodies, is infinite.
Fine discretizations: earlier results
While typical convex bodies are neither smooth objects, nor are they polyhedral surfaces, Zamfirescu’s result strongly suggests that equilibria in abundant numbers may occur in physically relevant scenarios. This is indeed the case if we study the surfaces of natural pebbles which, while rolling on a horizontal plane, are supported on their convex hull  and exhibit flocks of equilibria (cf. Figure 3).
In  we provided a mathematical justification for this observation. We studied the inverse phenomenon: namely, we were seeking the numbers and types of static equilibrium points of the families of polyhedra arising as equidistant -discretizations on an increasingly refined grid of a smooth curve with generic equilibrium points, denoted by ). In the planar case, as , and we find that the diameter of each of the flocks on (appearing around ) shrink and approach zero. However, we also find that inside a fixed domain (centered at ), the numbers of equilibria in each flock fluctuate around specific values and that are independent of the mesh size and the parametrization of the surface. We called these quantities the imaginary equilibrium indices associated with . We may eliminate the fluctuation of , by averaging over meshes in random positions (with uniform distributions) and in Theorem 1 of  we obtained for the planar case that
where , and is the (signed) curvature of at . What we did not prove was whether by summing over all imaginary equilibrium indices
actually provides all equilibria on the mesh. We will provide this result for the -dimensional case in Subsection 2.1 by proving Theorem 1 claiming that there exist no ‘irregular’ equilibria on the mesh which are separated from the flocks. In Subsection 2.2 we will prove Theorem 2 about randomized meshes indicating that these formulae are robust and they approximately predict results computed on non-uniform meshes.
2. Static theory: equilibria on finely discretized, fixed planar curves
Throughout this section, we deal with a planar curve satisfying the differentiability property which has exactly one, non-degenerate equilibrium point with respect a given reference point . Note that as a plane curve is a one-dimensional submanifold of , there is a neighborhood of in which the examined curve can be defined as a simple , three times continuously differentiable curve, where and . For the evolution of static equilibria we may write in a more explicit notation where is the spatial parametrization of and for planar curves is a scalar, for surfaces it is a vector. We will use the shorthand notation if it is clear from the context which function and which reference point are involved.
By a suitable choice of the coordinate system, we may assume that our reference point is the origin, and that is on the positive half of the -axis; i.e. . This implies that , and we also assume that . We restrict our investigation to curves that are ‘locally convex’ with respect to , and whose equilibrium point is non-degenerate. In other words, we assume that the signed curvature of at satisfies the inequalities .
Let denote the -segment equidistant partition of with . If is a division point of , then we introduce the notation , where . Note that the indices of the points are not necessarily integers, but real numbers that are congruent . We denote the set of indices of the points by , and examine the equilibria of the approximating curve . During our investigation, we assume that these equilibria are generic; that is, if has an equilibrium point at a vertex , then the vector is perpendicular to neither nor .
For any , let denote the number of stable equilibria of with respect to lying on the sides satisfying . We define the quantity for unstable equilibria of analogously. In Theorem 1 of , we proved that if is sufficiently large, then in the limit and will fluctuate around the so-called imaginary equilibrium indices given in equation (3).
Roughly speaking, we may say that these formulas hold for number of equilibria with ‘bounded’ indices, and we observe that this result holds for all sufficiently small , and the quantities in the formulas are independent of both and the parametrization of the curve.
2.1. Nonexistence of irregular equilibria
Whereas it is easy to see that the equilibria of are ‘physically’ close to (i.e. for any equilibrium point , is arbitrarily close to if is sufficiently small), this property does not imply that the set of indices of equilibrium points is bounded by some independent of . Thus, the formulas in (3) do not necessarily provide the numbers of all equilibrium points of . This led in  to the following definition:
If is a sequence of equilibrium points of with and , then the sequence is called an irregular equilibrium sequence.
In  it is remarked that if the coordinate functions of are polynomials, then the curve has no irregular equilibrium sequence, and asked (Question 1, ) whether the same holds if the two coordinate functions are analytic. Here we prove that the answer to this question is affirmative not only for every analytic, but for every -class curve. This means that the quantities in (3) are valid for all equilibrium points of , if is sufficiently small.
If is sufficiently small then there exist some values such that contains an equilibrium point if, and only if ; i.e. for sufficiently small the set of indices of equilibrium points is ‘connected’. In particular, the curve has no irregular equilibrium sequences.
We prove the assertion for the case that ; that is, the curve has a stable equilibrium at , for the case that we may apply a similar argument. For every (sufficiently small) , define the function by the implicit equation . We show that this function is strictly increasing, if is sufficiently small.
Clearly, . Thus, as is -class, there is some such that if , for some continuous vector function , we have
Consider the -variable function . If at a point , , then , since the angle of the two vectors is greater than if . Since all partials of are continuous, by the Implicit Function Theorem is continuously differentiable. Furthermore,
We have observed that for the denominator in this equation, we have for every , if . On the other hand, substituting (5) into the numerator, we obtain that
where is a continuous function. Note that if is sufficiently small. Thus, as if , we have if is sufficiently small, which yields that on this interval is strictly increasing. This proves the existence of . To show the existence of , we may apply the same consideration for the case that . ∎
2.2. Equilibria on random meshes
In this subsection we prove a probabilistic version of the formulae (3) which are the main planar result in , and deals with an equidistant, -element partition of the interval . Our goal is to show that even if the discretization is non-uniform, those formulae provide good estimates, thus the numbers predicted by the formulae may be observed in computer simulations.
Let be a -class curve satisfying the conditions in the beginning of Section 2. For arbitrary and define the probability distribution in the following way: Choose points independently and using uniform distribution on . Label the points such that , and set . Then is defined by
where . Then, for every there is some such that
By the Implicit Function Theorem, the coordinate function of is invertible in a neighborhood of , and thus, in this neighborhood the curve can be written as the graph of a function . Since the function and its inverse are both -class, a uniform distribution for on the interval correspond to an ‘almost’ uniform distribution of on the interval for sufficiently small values of . Thus, it suffices to prove the assertion for graphs of -variable functions, i.e. we may assume that is defined by for some -class function . Note that then .
Now, choose some values independently, and for , let . Observe that there is an equilibrium point on the segment if, and only if (see also (3) in ).
In the following, we use the second-degree Taylor polynomial of , by which we have
where is a continuously differentiable function on , implying that there is some such that for every . Then
Thus, if is sufficiently small, the sign of is ‘almost always’ equal to the sign of . We obtain similarly that the sign of is ‘almost always’ equal to that of .
Let denote the probability distribution where, choosing points uniformly and independently, is the probability that exactly pairs satisfy the inequalities
By our previous argument, it suffices to prove that . To do it, we distinguish four cases depending on the value of , and observe that convexity and the nondegeneracy of the equilibrium point implies that . These cases are , , , and . We note that the computations in all these cases are almost identical, and thus, we carry them out only in the first case.
Accordingly, assume that . We compute the probability that (6) is satisfied for some fixed value of . Putting , in this case the inequalities in (6) are equivalent to if , and if . Thus, is equal to the fraction of the volumes of two regions. The region in the denominator is the simplex defined by the inequalities ; its volume is equal to . The region in the numerator is equal the union of two nonoverlapping regions, which are defined by the inequalities
, , ,
, and .
By the linearity of expectation, we obtain that the expected value of the number of indices satisfying (6) is equal to . Summing up and applying the Binomial Theorem, we have
from which an elementary computation yields that
It is easy to check that the function is strictly decreasing on the interval , and that .
Finally, we show how one can reconstruct the number of equilibrium points of a smooth plane curve from one of its sufficiently fine discretizations. We state it in a slightly different form, for graphs of functions. In our setting, the function in Remark 4 is the Euclidean distance function of from the given reference point.
Let be a -class function with finitely many stationary points, each in the open interval , such that the second derivative of at each such point is not zero. Assume that has local minima and local maxima in . Let denote the division points of the equidistant -element partition of . Then, if is sufficiently large, there are exactly integers satisfying , and integers satisfying .
3. Dynamic theory: local equilibria on finely discretized, evolving planar curves
In this section, we deal with a -parameter () family of closed convex curves , where is time, and is the spatial parameter. We assume that is a -class function of , and this function, and all its derivatives with respect to , depend continuously on .
We denote the evolute of the function by . We say that a point of is general if it is not a cusp, and for any , . We say that is locally convex at a general point if has a neighborhood in such that is a strictly convex curve. If satisfies this property, the convex, connected region of is called the convex side of at , and the other region is called the concave side of .
Let be a -parameter family of convex curves satisfying the conditions above. Let be a continuous curve that transversely intersects at , and let (resp. ) denote the number of global equilibrium points (the sum of the imaginary equilibrium indices) of with respect to . Assume that is locally convex at the general point .
If moves from the convex side of to its concave side as increases, then increases by at , and .
If moves from the concave side of to its convex side as increases, then decreases by at , and .
As we remarked in the introduction, we call the events in (i) and in (ii) in the evolution of a -type and an -type event, respectively. The same events in the evolution of correspond to a generic, codimension one saddle-node bifurcation and they are called in bifurcation theorycreation and annihilation, respectively .
Let be a -class closed, convex curve. Let the evolute of the curve be . Assume that is locally convex at a point . Let denote the number of equilibrium points of with respect to . Then has a neighborhood such that for any point on the convex, and any point on the concave side of at .
Without loss of generality, throughout the proof we consider only spherical neighborhoods of . We recall the well-known fact that moving continuously, changes if, and only if crosses the evolute [17, 31]. Since for any sufficiently small neighborhood of , intersects in a simple curve, we have that for any , depends only on which side of is located. Thus, it suffices to prove that for some on the convex, and some on the concave side of , we have .
Let denote the normal line of at , i.e. the line perpendicular to and passing through . Note that the number of equilibria with respect to a point is the number of normal lines of passing through . Let the normal lines of passing through be . Let and be points sufficiently close to such that is on the convex and is on the concave side. Then, apart from some small neighborhood of , there are exactly normals (say , where ) passing through , and normals (say , ) passing through , where and are ‘close’ to . On the other hand, there are exactly two lines through that touch near , and since is a general point of , these two lines are normal lines of at exactly two values and , close to . Since every normal line of is tangent to , it follows that . Similarly, there are no lines through that touch close to , yielding that . ∎
For any fixed, small , the lines, normal to a side of the approximating polygon, and passing through a vertex of the side, decompose the plane into pieces of small diameters. The union of these normals, which we may call the evolute of the polygon, has the property that the number of the equilibria of the polygon changes if, and only if the reference point crosses this set . Thus, even though imaginary equilibrium indices change continuously during a continuous motion of the reference point, the quantity makes rapid jumps during this motion. This phenomenon can be observed, i.e. on Figure 4, especially when the reference point approaches the evolute of the curve.
Let the vertices of the approximating polygon be in counterclockwise order, and let us orient the two normals to the side at and such that the part of the normal through in the polygon points towards , and the part of the other normal in the polygon points away from . In this case, similarly like in Theorem 3, we can determine how the number of equilibria of the polygon changes if we cross its evolute: it increases if we cross an oriented line in it from right to left, and decreases in the opposite case. We note that the boundary of a cell in the cell decomposition defined by the evolute of the polygon is cyclic if, and only if the number of equilibria has a local extremum in this region.
3.1. The size of Critical Flocks
In this subsection we examine the number of local equilibria of the one-parameter family of curves at some fixed time , exactly when the reference point is on the evolute of the curve, so we will drop the subscript of . Strictly speaking, this subsection could also be regarded of the static theory of equilibria, discussed in Section 2, however, its content is more closely related to the subject of the current section. More specifically, we use the following notation.
Let be -class plane curve which has a unique equilibrium point with respect to the origin , where is the order of the first nonzero derivative of the Euclidean distance function at . As in Section 2, let be the -element equidistant partition of with . Let denote the number of equilibrium points of the approximating polygon defined by .
We have , i.e. there are constants such that holds for all .
The proof of Theorem 4 shows a slightly stronger statement, namely that the diameter of the flock at is of order , i.e. there are constants such that for any division point of with , both and the segment contains equilibrium points, and if , then neither.
Using the formula , we need to prove that . Let for any division point of . Note that if and are unstable equilibrium points, then contains a stable equilibrium, and if and contain stable equilibria, then is an unstable equilibrium. Thus, it suffices to prove the existence of constants such that if , then contains a stable equilibrium point, and if , then it does not. Clearly, in the proof we may assume that is sufficiently large.
We prove Theorem 4 only under the additional assumption that denotes the polar angle in a suitable polar coordinate system. We remark that in the general case, in a suitable coordinate system, the polar angle can be expressed as a -class function of in a neighborhood of , and in a small neighborhood of , we have for some suitable . Using this inequality, our argument can be modified for a general parametrization of in a straightforward way. Under the assumption that denotes polar angle, the curve can be written as , where is the -class positive distance function .
Similarly as before, we observe that there is a stable point on if, and only if . In our notation, these inequalities are equivalent to
First, by the differentiability properties of , there are constants and such that
We carry out the computations only for positive indices , and assuming that ; that is, assuming that for small positive values of , is a decreasing function. Under these conditions, the second inequality is satisfied for all values of .
Assume now that the first inequality is satisfied. Using the inequalities , we obtain that then
Now, let us estimate the right-hand side from above by (8). Then, applying the inequalities for all real and integer , we obtain that
where . Let , where will be chosen later. Then an elementary consideration shows that the largest member of the above expression, in terms of , is of order , and by the inequality , we have that if is sufficiently small, then